Alternating-gradient Focusing Research Articles

Table-top ion-trap experiment on the stability of intense short bunches in linear hadron accelerators

The novel experimental system “S-POD” (Simulator of Particle Orbit Dynamics) is employed to explore the stability of short hadron bunches in high-intensity linacs. In a previous study with the S-POD [M. Goto et al., ], a static potential was used to focus the bunch in the longitudinal direction. We here make a step forward to include the possibility of pure synchrotron resonance, introducing periodic modulation to the longitudinal potential well. The modulation period was taken a half of the transverse alternating-gradient focusing period, which reflects the most typical lattice condition of a drift-tube linac. Detailed stability maps are constructed to reveal dangerous parameter regions where serious beam loss may occur due to resonance. We reconfirm the existence of various betatron and synchrobetatron resonance stop bands whose widths and locations change in tune space depending on the bunch intensity. It turns out that the periodicity of the longitudinal focusing potential brings about no pronounced effect on the resonance feature; the result is very similar to what we obtained in the previous study with a static longitudinal potential. As long as the lattice periodicity mentioned above is maintained, no serious noncoupling synchrotron resonance appears even with a high synchrotron phase advance above 90° per unit alternating-gradient cell. Severe envelope instability may, however, be excited in the longitudinal direction if the axial focusing force includes error components that affect the original lattice periodicity. The experimental observations can be explained with the free from the concept of incoherent tune spread. Published by the American Physical Society 2024

Mathieu unit cell as a template for low emittance lattices

The multi-bend achromat (MBA), which often serves as a building block for modern low-emittance storage rings, is composed of a repetition of unit cells with optimized optical functions for low emittance in the achromat center, as well as end cells for dispersion and optics matching to insertion devices. In this work, we describe the simplest stable class of unit cells that are based on a longitudinal Fourier expansion, transforming Hill equations to Mathieu equations. The resulting cell class exhibits continuously changing dipolar and quadrupolar moments along the beam path. Although this elementary model is defined by only three parameters, it captures a significant amount of notions that are applied in the design of MBAs. This is especially interesting as Mathieu cells can be viewed as an elementary extension of Christofilos' original model of alternating-gradient focusing, while their sinusoidal bending and focusing functions lend themselves to future applications in undulator-like structures. Mathieu cells can be used to estimate the range of reasonable cell tunes and put an emphasis on the combination of longitudinal gradient bending and reverse bending, as well as on strong horizontal focusing to reach emittances lower than the classic theoretical minimum emittance cell. Furthermore, the lowest emittances in this model are accompanied by small absolute momentum compaction factors.

Experimental observation of low-order collective oscillation modes in a strong-focusing lattice

In a conventional linear Paul trap (LPT), four electrode rods are placed symmetrically around the trap axis to generate a radio-frequency (rf) quadrupole field for transverse ion confinement. The periodic nature of the external focusing potential can give rise to serious ion losses under a specific condition. The loss mechanism is essentially the same as the coherent betatron resonance well-known in intense beam dynamics [1, 2]. In fact, the collective motion of an ion plasma in the LPT is shown equivalent to that of a charged-particle beam traveling through an alternating-gradient (AG) focusing lattice. In the present study, we perform the direct measurement of low-order coherent oscillation modes in the LPT by detecting image currents induced on the electrodes' surfaces. The four-rod structure of the LPT allows us to pick up weak signals from the dipole and quadrupole oscillations of a plasma bunch. These signals are Fourier analyzed to evaluate the coherent oscillation tune at different initial ion densities. The measured tune of the quadrupole mode is used to deduce the tune depression as a function of ion number stored in the LPT.

Structure resonances due to space charge in periodic focusing channels

The Vlasov-Poisson model is one of the most effective methods to study the space charge dominated beam evolution self-consistently in a periodic focusing channel. Since the approach to get the solution with this model is not trivial, previous studies are limited in degenerated conditions, either in smoothed channel (constant focusing) [I. Hofmann, Phys. Rev. E 57, 4713 (1998)] or in alternating gradient focusing channel with equal initial beam emittance condition in the degrees of freedom [I. Hofmann et al., Part. Accel. 13, 145 (1983); Chao Li et al., THOBA02, IPAC2016]. To establish a basis, we intentionally limit this article to the study of the pure transverse periodic focusing lattice with arbitrary initial beam condition, and the same lattice structure in both degrees of freedom, but with possibility of different focusing strengths. This will show the extension of the existing work. The full Hamiltonian is invoked for a pure transverse focusing lattice in various initial beam conditions, revealing different mode structure and additional modes beyond those of the degenerated cases. Application of the extended method to realistic lattices (including longitudinal accelerating elements) and further details will then reveal many new insights, and will be presented in later work.

Near equilibrium distributions for beams with space charge in linear and nonlinear periodic focusing systems

A procedure to obtain a near equilibrium phase space distribution function has been derived for beams with space charge effects in a generalized periodic focusing transport channel. The method utilizes the Lie transform perturbation theory to canonically transform to slowly oscillating phase space coordinates. The procedure results in transforming the periodic focusing system to a constant focusing one, where equilibrium distributions can be found. Transforming back to the original phase space coordinates yields an equilibrium distribution function corresponding to a constant focusing system along with perturbations resulting from the periodicity in the focusing. Examples used here include linear and nonlinear alternating gradient focusing systems. It is shown that the nonlinear focusing components can be chosen such that the system is close to integrability. The equilibrium distribution functions are numerically calculated, and their properties associated with the corresponding focusing system are discussed.

Design of a <formula formulatype="inline"><tex Notation="TeX">$\hbox{MgB}_{2}$</tex></formula> Beam Transport Channel for a Strong-Focusing Cyclotron

A superconducting strong-focusing cyclotron is being developed for high current applications. Alternating-gradient focusing is provided by ~ 6 T/m superconducting beam transport channels which lie in the sectors along the arced beam trajectory of each orbit of the cyclotron. The ~ 1 T sector dipoles, corrector dipoles, and Panofsky type quadrupoles utilize MgB <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> superconductor operating around 15-20 K. The operating temperature provides a valuable margin for a cost-effective cryogenic design, and large thermal stability in the event of occasional heat loads from intercepted beam or other sources. The main dipole windings are designed with sufficiently large curvature so that they can be fabricated using react-and-wind procedure; the quadrupole windings require small-radius end bends and so must be fabricated using wind-and-react procedure. Initial magnetic modeling on the end field region is presented.

Effect of image charge on the off-axis transport of intense beams in a small aperture quadrupole lattice

This paper investigates the first order image charge effects of circular conducting pipe on an intense off-axis continuous charged particle beam propagating through an alternating-gradient focusing channel. The equations of motion for the centroid and envelopes of slightly off-axis beam are derived and used to determine the equilibrium and stability conditions for the beam transport. The envelope oscillation frequencies of the mismatched beam are obtained by the smooth approximation method and by numerical evaluation of the linearly perturbed envelope equations. Depending on the beam intensity and focusing parameters, it is found that the frequency of the out-of-phase mode of envelope oscillation is reduced due to the presence of the conducting wall. Detailed numerical results for off-axis beam transport in a quadrupole lattice are presented and the stable regimes in the parameter space are determined numerically. It is observed that the unstable region for the envelope evolution with image charge effects depends on beam intensity and external focusing strength and increases with the off centering of the beam.

Experimental simulation of beam propagation over long path lengths using radio-frequency and magnetic traps

An overview is given of the novel beam-dynamics experiments based on compact non-neutral plasma traps at Hiroshima University. We have designed and constructed two different classes of trap systems, one of which uses a radio-frequency electric field (Paul trap) and the other uses an axial magnetic field (Penning trap) for transverse plasma confinement. These systems are called “S-POD” (Simulator for Particle Orbit Dynamics). The S-POD systems can approximately reproduce the collective motion of a charged-particle beam propagating through long alternating-gradient (AG) quadrupole focusing channels using the Paul trap and long continuous focusing channels using the Penning trap. This allows us to study various beam-dynamics issues in compact and inexpensive experiments without relying on large-scale accelerators. So far, the linear Paul traps have been applied for the study of resonance-related issues including coherent-resonance-induced stop bands and their dependence on AG lattice structures, resonance crossing in fixed-field AG accelerators, ultralow-emittance beam stability, etc. The Penning trap with multi-ring electrodes has been employed primarily for the study of beam halo formation driven by initial distribution perturbations. In this paper, we briefly overview the S-POD systems, and then summarize recent experimental results on resonance effects and halo formation.

Adiabatic formation of a matched-beam distribution for an alternating-gradient quadrupole lattice

The formation of a quasiequilibrium beam distribution matched to an alternating-gradient quadrupole focusing lattice by means of the adiabatic turn-on of the oscillating focusing field is studied numerically using particle-in-cell simulations. Quiescent beam propagation over several hundred lattice periods is demonstrated for a broad range of beam intensities and vacuum phase advances describing the strength of the oscillating focusing field. Properties of the matched-beam distribution are investigated. In particular, self-similar evolution of the beam density profile is observed over a wide range of system parameters. The numerical simulations are performed using the WARP particle-in-cell code.

Experimental investigation of random noise-induced beam degradation in high-intensity accelerators using a linear Paul trap

A random noise-induced beam degradation that could affect intense beam transport over long propagation distances has been experimentally investigated by making use of the transverse beam dynamics equivalence between an alternating-gradient focusing system and a linear Paul trap system. For the present study, machine imperfections in the quadrupole focusing lattice are considered, which are emulated by adding small random noise on the voltage waveform of the quadrupole electrodes in the Paul trap. It is observed that externally driven noise continuously increases the rms radius, transverse emittance, and nonthermal tail of the trapped charge bunch almost linearly with the duration of the noise. The combined effects of collective modes and colored noise are also investigated and compared with numerical simulations.

Use of a Linear Paul Trap to Study Random Noise-Induced Beam Degradation in High-Intensity Accelerators

A random noise-induced beam degradation that can affect intense beam transport over long propagation distances has been experimentally studied by making use of the transverse beam dynamics equivalence between an alternating-gradient (AG) focusing system and a linear Paul trap system. For the present studies, machine imperfections in the quadrupole focusing lattice are considered, which are emulated by adding small random noise on the voltage waveform of the quadrupole electrodes in the Paul trap. It is observed that externally driven noise continuously produces a nonthermal tail of trapped ions, and increases the transverse emittance almost linearly with the duration of the noise.

Adiabatic warm-fluid equilibrium theory of thermal charged-particle beams in alternating-gradient focusing fields

An adiabatic warm-fluid equilibrium theory for a thermal charged-particle beam in an alternating-gradient focusing field is presented. Warm-fluid equilibrium equations are solved in the paraxial approximation. The theory predicts that the four-dimensional rms thermal emittance of the beam is conserved, but the two-dimensional rms thermal emittances are not constant. The rms beam envelope equations and the self-consistent Poisson equation, governing the beam density and potential distributions, are derived. Although the presented rms beam envelope equations have the same form as the previously known rms beam envelope equations, the evolution of the rms emittances in the present theory is given by analytical expressions. The density does not have the simplest elliptical symmetry, but the constant-density contours are ellipses, and the aspect ratio of the elliptical constant-density contours decreases as the density decreases along the transverse displacement from the beam axis. For high-intensity beams, the beam density profile is flat in the center of the beam and falls off rapidly within a few Debye lengths, and the rate at which the density falls is approximately isotropic in the transverse directions.

Cosine gradient theory in cyclotrons with general dees

This paper gives the representation of the electric field for cyclotron accelerating gap with general dee geometry. A special case of this representation is the cosine gradient theory of central region electric field developed by Kost and Mackenzie [IEEE NS-22 (3) (1975) 1922] in TRIUMF for 180° dees. A general expression of energy gain and a formula to determine the vertical electric focusing, which improves the alternating-gradient focusing proposed by Gordon and Marti [Part. Acceler. 11 (1980) 161] using a simple electric-field model, are obtained. In our theory, the velocity-variation π / 2 focusing and radius-change effect have been predicted to perfect the calculation of the vertical electric focusing. We take the 100 MeV cyclotron under design and construction at China Institute of Atomic Energy, CYCIAE-100 [T. Zhang, X. Guan et al., in: Proceedings of the Third Asia Particle Accelerator Conference, Korea, 2004, p. 267], as an application of this theory in two-dee cyclotron running at push-push mode.

Stark deceleration of OH radicals in low-field-seeking and high-field-seeking quantum states

The Stark deceleration of OH radicals in both low-field-seeking and high-field-seeking levels of the rovibronic $^{2}\ensuremath{\Pi}_{3∕2}$, $v=0$, $J=3∕2$ ground state is demonstrated using a single experimental setup. Applying alternating-gradient focusing, OH radicals in their low-field-seeking $^{2}\ensuremath{\Pi}_{3∕2}$, $v=0$, $J=3∕2$, $f$ state have been decelerated from $345\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}239\phantom{\rule{0.3em}{0ex}}\mathrm{m}∕\mathrm{s}$, removing 50% of the kinetic energy using only 27 deceleration stages. The alternating-gradient decelerator allows one to independently control longitudinal and transverse manipulation of the molecules. Optimized high-voltage switching sequences for the alternating-gradient deceleration are applied, in order to adjust the dynamic focusing strength in every deceleration stage to the changing velocity over the deceleration process. In addition we have also decelerated OH radicals in their high-field-seeking $^{2}\ensuremath{\Pi}_{3∕2}$, $v=0$, $J=3∕2$, $e$ state from $355\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}316\phantom{\rule{0.3em}{0ex}}\mathrm{m}∕\mathrm{s}$. For the states involved, a real crossing of hyperfine levels occurs at $640\phantom{\rule{0.3em}{0ex}}\mathrm{V}∕\mathrm{cm}$, which is examined by varying a bias voltage applied to the electrodes.

Alternating-gradient focusing and deceleration of large molecules

We have focused and decelerated benzonitrile $({\mathrm{C}}_{7}{\mathrm{H}}_{5}\mathrm{N})$ molecules from a molecular beam, using an array of time-varying inhomogeneous electric fields in alternating-gradient configuration. Benzonitrile is prototypical for large asymmetric top molecules that exhibit rich rotational structure and a high density of states. At the rotational temperature of $3.5\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ in the pulsed molecular beam, many rotational states are populated. Benzonitrile molecules in their absolute ground state are decelerated from $320\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}289\phantom{\rule{0.3em}{0ex}}\mathrm{m}∕\mathrm{s}$, and similar changes in velocity are obtained for excited rotational states. All measurements agree well with the outcome of trajectory calculations. These experiments demonstrate that such large polyatomic molecules are amenable to the powerful method of Stark deceleration.

EMITTANCE LIMITATION OF A CONDITIONED BEAM IN A STRONG FOCUSING FEL UNDULATOR

Various methods have been proposed to condition an electron beam in order to reduce its emittance effect and to improve the short-wavelength free electron laser (FEL) performance. In this paper, we show that beam conditioning does not result in a complete elimination of the emittance effect in an alternating-gradient focusing FEL undulator. Using a one-dimensional model and a three-dimensional simulation code, we derive a criteria for the emittance limitation of a perfectly conditioned beam that depends on the focusing structure.

Experiments on transverse compression of a long charge bunch in a linear Paul trap

The transverse compression of a long charge bunch is investigated in the Paul trap simulator experiment (PTSX), which is a linear Paul trap that simulates the nonlinear transverse dynamics of an intense charged particle beam propagating through an equivalent kilometers-long magnetic alternating-gradient (AG) focusing system. Changing the voltage amplitude at fixed focusing frequency in the PTSX device corresponds to changing the field gradient of the quadrupole magnets with fixed axial periodicity in the AG transport system. In this work, we present experimental results on transverse compression of the charge bunch in which the amplitude of the applied oscillatory focusing voltage is changed instantaneously, and adiabatically. The experimental data are also compared with analytical estimates and 2D WARP particle-in-cell simulations.

Parametric studies of image-charge effects on an intense beam in a small-aperture alternating-gradient focusing system

The image-charge effects on an intense charged-particle beam propagating through an alternating-gradient focusing channel with a small aperture, circular, perfectly conducting pipe are studied using a test-particle model. For a well-matched elliptical beam with the Kapchinskij–Vladimirskij (KV) distribution, it is found that halo formation and beam loss are induced by nonlinear fields due to image charges on the wall. The halo formation and chaotic particle motion is found to depend sensitively on the system parameters: the occupancy of the quadrupole focusing field, the vacuum phase advance, the beam perveance, and the ratio of the beam size to the aperture. Furthermore, the percentage of beam loss to the conductor wall is calculated as a function of propagating distance and aperture.

Experimental study of the transport limits of intense heavy ion beams in the HCX

The High Current Experiment (HCX) at Lawrence Berkeley National Laboratory is part of the US program to explore heavy-ion beam transport at a scale representative of the low-energy end of an induction linac driver for fusion energy production. The primary mission of this experiment is to investigate aperture fill factors acceptable for the transport of space-charge-dominated heavy-ion beams at high space-charge intensity (line charge density up to ∼0.2 μC/m) over long pulse durations (4 μs) in alternating gradient focusing lattices of electrostatic or magnetic quadrupoles. The experiment also contributes to the practical baseline knowledge of intense beam manipulations necessary for the design, construction and operation of a heavy ion driver for inertial fusion. This experiment tests transport issues resulting from nonlinear space-charge effects and collective modes, beam centroid alignment and beam steering, matching, image charges, halo, electron cloud effects, and longitudinal bunch control. We first present the results for a coasting 1 MeV K + ion beam transported through the first ten electrostatic transport quadrupoles, measured with optical beam-imaging and double-slit phase-space diagnostics. This includes studies at two different radial fill factors (60% and 80%), for which the beam transverse distribution was characterized in detail. Additionally, beam energy measurements will be shown. We then discuss the first results of beam transport through four pulsed room-temperature magnetic quadrupoles (located downstream of the electrostatic quadrupoles), where the beam dynamics become more sensitive to the presence of secondary electrons.

Chaotic particle motion and beam halo formation induced by image-charge effects in a small-aperture alternating-gradient focusing system

It is shown with a test-particle model that image-charge effects induce a new mechanism for chaotic particle motion and halo formation in an intense charged-particle beam propagating through an alternating-gradient focusing channel with a small aperture, circular, perfectly conducting pipe. This mechanism occurs for a well-matched beam with the Kapchinskij–Vladimirskij distribution. The percentage of beam loss to the conductor wall is calculated as a function of propagating distance and aperture size.